ZENO’S ARGUMENTS ABOUT MOTION 281 
attempt to extend to the values ot' a variable the variability 
which belongs to it alone. When once it is firmly realized 
that all the values of a variable are constants, it becomes easy 
to see, by taking any two such values, that their difference is 
always finite, and hence that there are no infinitesimal differ 
ences. If x be a variable which may take all real values 
from 0 to 1, then, taking any two of these values, we see that 
their difference is finite, although a? is a continuous variable. 
It is true the difference might have been less than the one we 
chose; but if it had been, it would still have been finite. The 
lower limit to possible differences is zero, but all possible 
differences are finite; and in this there is no shadow of 
contradiction. This static theory of the variable is due to the 
mathematicians, and its absence in Zeno’s day led him to 
suppose that continuous change was impossible without a state 
of change, which involves infinitesimals and the contradiction 
of a body’s being where it is not.’ 
In his later chapter on Motion Mr. Russell concludes as 
follows: 1 
‘ It is to be observed that, in consequence of the denial 
of the infinitesimal and in consequence of the allied purely 
technical view of the derivative of a function, we must 
entirely reject the notion of a state of motion. Motion consists 
merely in the occupation of different places at different times, 
subject to continuity as explained in Part V. There is no 
transition from place to place, no consecutive moment or 
consecutive position, no such thing as velocity except in the 
sense of a real number which is the limit of a certain set 
of quotients. The rejection of velocity and acceleration as 
physical facts (i. e. as properties belonging at each instant to 
a moving point, and not merely real numbers expressing limits 
of certain ratios) involves, as we shall see, some difficulties 
in the statement of the laws of motion; but the reform 
introduced by Weierstrass in the infinitesimal calculus has 
rendered this rejection imperative.’ 
We come lastly to the fourth argument (the Stadium). 
Aristotle„’s representation of it is obscure through its extreme 
brevity of expression, and the matter is further perplexed by 
an uncertainty of reading. But the meaning intended to be 
conveyed is fairly clear. The eight M’s, B’s and G’s being 
1 Op. clt., p. 478.